#1 -------------------

"Harry-Potter"
For the Triwizard Tournament 60 of 280 students have reached the minimum age of 17 to apply to the tournament.
10 of the students above full age (18 y/o or older) have applied with a note with their names in the goblet of fire. But also 15 of the underaged students are trying to outwit the age limit of the goblet of fire.

1) You observe a student trying to put a note with his name in the goblet of fire. What is the probability that this student is underaged?

2) You met a student, who did not put a note with his name in the goblet of fire. What is the probability that this student is of full age?

#2 -------------------
"Problem of Points / Luca Pacioli"
2 Players play a gambling with multiple rounds. Each player pay the same stake from the beginning. Each round one player wins.
Every round will be played independently. The probability of winning for both players is 50%.
The player who win 10 rounds first is winning the whole game and receiving the stakes.
The game needs to be stopped because of a fire.
The current score is 7:8

Question: How does the stake needs to be divided/split to the both players so that the chance of win of both players is correctly protrayed.

#3 -------------------
A fair coin is flipped 3 times. The events A,B,C,D are defined:

A := {in 1st flip head will win}
B := {minimum 2 times tail will be flipped}
C := {2nd flip is head}
D := {in 1st and 2nd flip different events will occur}

1) Show that P(A∩B∩C)=P(A)P(B)P(C) but A,B,C are not independent
2) Show the A,C,D are pairly independent but not independent

#4 -------------------
FYI: Pot = Power set
Let (Ω,P) a probability space and A,B,C ∈ Pot(Ω)
1) Let P(N)∈(0,1). Show that A,B, are independent if P(A|B)=P(A|B^c)
2) A,B,C are independent. Show that also A∪B,C are independent.